Methodology and Algorithms for Protecting Centrifugal and Axial Compressors from Surge and Choke

ABSTRACT

This disclosure describes a novel methodology for anti-surge and anti-choke control systems protecting centrifugal and axial compressors. The methodology, based on Buckingham&#39;s π-theorem for compressors, presents compressor performance maps in dimensionless rectangular π-term coordinates that are independent of compressor inlet conditions, fluid molecular weight and rotational speed. The full range of compressor operating points from surge to choke is monitored and controlled when surge and choke limits are available. This is accomplished by converting rectangular coordinates presented in π-terms to polar coordinates, and then converting them to a controlled variable used in the closed-loop controllers. The methodology provides control algorithms for variable speed compressors, variable geometry compressors equipped with inlet guide vanes or stator vanes that exhibit displacement of surge and choke limits. The methodology most accurately estimates the location of the operating point relative to its limit in polar coordinates if only the surge or choke limit is available. The presented protection methods are applicable to any known types of dynamic compressors for industrial, commercial, jet engines, turbochargers.

TECHNICAL FIELD

The present invention generally relates to methods for protectingdynamic compressors from surge and choke using controlled variablesobtained from converted compressor performance maps provided bycompressor manufacturers or experimentally obtained during commissioningfor use in control systems. More specifically, it relates to methodsthat most accurately estimates the operating point position relative tosurge and choke limits by polar conversion to ensure most efficientcompressor operation using conventional PID control.

BACKGROUND ART

There are two restrictions on the operation of centrifugal and axialcompressors caused by different phenomena: at relatively low flowrates—surge; and at relatively high flow rates, a choke or a stone wall.These conditions must be taken into account in the design so that theycan be prevented. Close-loop proportional-integral-derivativecontrollers are commonly used to regulate anti-surge valves to protectcompressors from surge, and to regulate IGV inlet guide vanes or outletvalves to prevent compressor choking.

Both surge and choke are fluid mechanics phenomena that occur incompressors under certain circumstances. Then the theory of fluidmechanics is the basis for ensuring dynamically stable compressoroperation. Fluid mechanics problems are too complex to be solvedanalytically, so their behavior must be verified experimentally. Sincedynamically unstable compressor operation due to surge or rapidefficiency drops due to choking can occur at different rotationalspeeds, inlet pressures, inlet temperatures and different molecularweights of fluids (in industrial applications), the principle ofsimilarity and dimensional analysis are used. The goal of similarity anddimensional analysis is to reduce the number and complexity of processparameters that affect a given physical phenomenon, in the case ofcompressors, the compressor performance can be displayed indimensionless rectangular coordinates, chosen so that the processparameters at the compressor inlet, such as pressure, temperature andgas composition, are irrelevant. In other words, any particularoperating point on the dimensionless compressor map defines thecorresponding prototypes regardless of inlet conditions, gascomposition, and even rotational speed. To find such dimensionlesscoordinates for compressors, the implicit functional relationship of allparameters involved in operation of the compressors can be analyzed inaccordance with Buckingham's π-theorem. The first most common π-termcharacterizing the compressor performance is the Mach number of theincoming or outgoing flow, depending on the location of the flow meter,which is very often replaced by the term “corrected mass flow rate” usedas the horizontal coordinate on compressor maps. The second most commonthe π-term is the ratio of the total outlet pressure to the total inletpressure used as the vertical coordinate on the compressor maps.

Since the axial velocity of the continuous flow of fluid entering orleaving the compressor is involved in the Mach number calculations, thechoice of flow meter is of great importance. Microturbines, vortexmeters, acoustic flow meters and other devices, that generate signalsproportional to the axial velocity of the fluid, can be used inapplications where the molecular weight of the fluid does not change.Differential pressure flow meters such as orifices. Venturi tubes,Venturi nozzles, annubars and etc. are required for applications wherethe molecular weight of the fluid changes and there is no molecularweight measurement.

There are various compressor protection algorithms for determining theposition of the operating point in relation to the surge or choke line,represented in dimensionless rectangular coordinates. However, none ofthese rectangular coordinate methods can simultaneously determine theposition of the operating point relative to the surge and choke lines inorder to reproduce the full operating range, defined as the continuousvalue from the surge limit to the choke limit. Compressor performancecurves in the plane are geometrically limited to perpendicularlyoriented lines to reproduce such a range in rectangular coordinates. Inaddition, compressor operation can be described as moving the operatingpoint towards surge or choke limits along the performance curves, orfrom curve to curve in the terms of a radius from some imaginary centerpoint. Such movements are more like movements in polar coordinates thanin rectangular ones. Therefore, polar coordinates seem to be the mostappropriate choice in a context where the operating point in question isinherently tied to a direction and length from a center point on aplane. To demonstrate significant progress over existing compressorprotection methods, the entire area between the surge and choke limitinglines must be converted from rectangular coordinates to polarcoordinates. This invention describes the procedure for such aconversion.

A polar coordinate system in a plane consists of a fixed center point ofthe pole or zero point and rays emanating from that point. In the polarcoordinate system selected in present invention, each point on a planehas a pair of polar coordinates: the radial coordinate r is the distancebetween the pole and the designated point, and φ (or α, or γ) is theangular coordinate, measured as the polar angle from the vertical axisto the radial coordinate r.

To convert a constant speed performance curve from a rectangularcoordinate system to a polar coordinate system, it is necessary to makethe assumption that each point on the performance curve in the newcoordinate system is approximately the same distance from the centerpoint. Or at least the radial coordinate of the surge point and theradial coordinate of the choke point, defined as the intersection pointsof the performance curve with the surge and choke lines, are the same.

However, a compressor control system usingproportional-integral-derivative controllers requires further conversionof the two-dimensional representation into a numeric string of theone-dimensional controlled variable CV. In general, if the fulloperating point range from surge limit to choke limit is specified, thenthe controlled variable CV (%) in percent for the surge protectioncontroller can be calculated relative to the surge limit from theequation below:

$\begin{matrix}{{{CV}(\%)} = {100{\% \cdot \frac{{CV} - {CV}_{{surge}\_{limit}}}{{CV}_{{choke}\_{limit}} - {CV}_{{surge}\_{limit}}}}}} & (1)\end{matrix}$

For the controller protecting the compressor from choking, thecontrolled variable CV (%) in percent is calculated relative to thechoke limit:

$\begin{matrix}{{{CV}(\%)} = {100 \cdot \frac{{CV}_{{choke}\_{limit}} - {CV}}{{CV}_{{choke}\_{limit}} - {CV}_{{surge}\_{limit}}}}} & (2)\end{matrix}$

The close-loop PID controller continuously calculates the error value ERas the difference between the desired setpoint SP (%) in percent and theinput value of the controlled variable CV (%) in percent to tune thecontrol output:

ER=SP (%)−CV (%)  (3)

For an anti-surge controller, the desired SP (surge protection margin)is usually 10% or less.

For an anti-choke controller, the desired SP (choke protection margin)is usually 5% or more.

In a case of the polar coordinate system, the controlled, variable CVbecomes the polar angle φ of the operating point with respect to itsradial coordinate r. The controlled variable CV (%) for a surgeprotection controller is calculated relative to the surge unlit at theconstant radial coordinate r as the polar angle φ of the operating pointminus the polar angle of the surge point φ_(surge), divided by the fullrange, defined as subtracting the polar angle of the surge point toφ_(surge) from the polar angle of the choke point φ_(choke):

$\begin{matrix}{{{CV}(\%)} = {100{\% \cdot \frac{\varphi - \varphi_{surge}}{\varphi_{choke} - \varphi_{surge}}}❘_{r}}} & (4)\end{matrix}$

The controlled variable CV (%) for a choke protection controller can becalculated relative to the choke limit for at the constant radialcoordinate r as the polar angle of the choke point minus the polar angleof the operating point, divided by the full range, defined assubtracting the polar angle of the surge point from the polar angle ofthe choke point:

$\begin{matrix}{{{CV}(\%)} = {100{\% \cdot \frac{\varphi_{choke} - \varphi}{\varphi_{choke} - \varphi_{surge}}}❘_{r}}} & (5)\end{matrix}$

Variable geometry compressors equipped with the IGV inlet guide vanes orstator vanes in axial compressors may exhibit surge and choke linesshift in response to blades opening. Since this displacement is stillexpressed in π-term coordinates, an IGV correction function can beapplied to the π-term coordinate of the Mach number as a function of theposition of the input guide vanes before converting the rectangularcoordinates to polar coordinates. The controlled variable CV (%) canthen be calculated.

Most industrial compressors are selected to operate at or near maximumefficiency. For this reason, compressor maps often have surge lines andno choke lines, but instead have maximum flow endpoints on constantspeed performance curves. Compressor testing to determine choke pointsduring commissioning may be process limited. In such, cases, where thefull range of compressor operation, defined from surge limit to chokelimit, is not available, only compressor surge protection is required.To this end, each surge point on the surge line assumes a constant polarangle. Assigning a constant polar angle to the surge points, requiresadjusting one of the two π-term coordinates, for instance, the Machnumber becomes a function of the π-term coordinate of the pressureratio, or vice versa, before converting the rectangular coordinates topolar coordinates. In the case of maximum flow endpoints, the controlledvariable CV (%) for the surge protection controller can be calculatedrelative to the surge limit as the polar angle of the operating pointminus the constant (as polar angle of the surge points), divided by thespecified operating range, defined as subtracting the constant from thepolar angle of the maximum flow endpoint.

The controlled variable CV (%), when only surge points collected duringcommissioning are available and no compressor performance maps arepresented, can be calculated for the surge protection controllerrelative to the surge limit as the polar angle of the operating pointminus a constant that defines the polar angle of the surge pointsdivided by this constant.

It is important to note that selecting 10% as the desired setpoint whenonly surge limits are present will denote a safety margin relative tothe surge limit and cannot be equal to the safety margin as a percentageof the compressor's full operating range.

The conversion methods described are applicable to any shape ofcompressor performance curves, from a slight slope of an almosthorizontal line to relatively straight vertical lines.

SUMMARY OF THE INVENTION

The present invention proposes novel algorithms for practical use incontrol systems that protect dynamic compressors from surging andchoking.

The rectangular to polar conversion reproduces the entire range ofoperating points with the most accurate positioning of the operatingpoints relative to surge and choke limits, effectively protecting thecompressors from both surge and choke.

The invention provides the most realistic representation of thecompressor operation in a wide range, regardless of changes in inputconditions, molecular weight, rotation speed, position of the imidevanes.

The new surge and choke protection algorithms presented in theinvention, when only surge points or only choke points are present, arethe most understood in compressor control practice.

The invention proposes new algorithms for calculating controlledvariables CV (%) that are nearly linear, which makes the tuning of thePID controllers very precise.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically shows the input and outlet signals; gas propertiesat the compressor inlet.

FIG. 2 shows the different shapes of the compressor performance curves.

FIG. 3 depicts the operating point in the π-term coordinates between thesurge and choke limits on a constant speed line, with rays emanatingfrom the pole indicating possible movements.

FIG. 4 shows a set, of hypothetical constant speed performance curveswith operating point, surge and choke limits.

FIG. 5 depicts the effect of transformation by plotting polarcoordinates r and φ on the rectangular coordinates.

FIG. 6 represents the correlation between the controlled variable CV (%)and the polytropic efficiency of the compressor.

FIG. 7 schematically shows two dynamic compressors 14 and 15 in series.

FIG. 8 shows three sets of four hypothetical constant speed performancecurves for three different IGV positions.

FIG. 9 describes the IGV function.

FIG. 10 reflects the effect of transforming the horizontal axis usingthe IGV function.

FIG. 11 depicts the effect of transformation by plotting polarcoordinates r and φ on the rectangular coordinates for variable geometrycompressors.

FIG. 12 shows a set of hypothetical constant speed performance curveswith operating point, surge line, and maximum flow points with raysemanating from the pole indicating the polar coordinate α of the surgepoints.

FIG. 13 represents the modified compressor map with a constant angularcoordinate α.

FIG. 14 depicts the effect of transforming by plotting polar coordinatesr and α on rectangular coordinates with only surge limit available.

FIG. 15 shows the most common surge line shape in rectangularcoordinates when the horizontal axis is the π-term Mach number at outletof the compressor.

FIG. 16 depicts a modified surge line in polar coordinates with aconstant angular coordinate γ.

FIG. 17 shows a set of hypothetical performance curves for a constantspeed, variable geometry compressor with surge line and maximum flowendpoints.

FIG. 18 shows the effect of transforming the horizontal axis using theIGV function and the rays emanating from the pole, indicating the polarcoordinate α of the surge points.

FIG. 19 shows a modified surge line with constant polar coordinate a forvariable geometry compressors.

FIG. 20 depicts the effect of transformation by plotting polarcoordinates r and α on rectangular coordinates for variable geometrycompressors for which only surge limit is available.

DESCRIPTION OF INVENTION

FIG. 1 shows a schematic diagram of dynamic compressor 1 and most commoninput signals: measured flow rate 2, static inlet pressure 3, staticinlet temperature 4, static output pressure 5, static output temperature6, rotational speed 7, position of the inlet guide vanes or stator vanes8; fluid properties 9, 10, 11; calculated axial fluid velocities 12, 13used in control algorithms.

There are a number of dimensionless groups (π-terms) that can beobtained from Buckingham's π-theorem applied to compressors, but themost commonly chosen π-terms are Mach number Π₁, and compressor totalpressure ratio Π₂. Both of these π-terms are used in present invention.The performance of dynamic compressors may be described by followingquantities:

-   m Fluid mass flow-   N Rotor rotational speed, usually measured as revolution per minute    (RPM)-   V Axial fluid velocity at the compressor inlet or outlet depending    on the location of the flow meter-   α The speed of sound at the inlet or outlet of the compressor-   Mw Fluid molecular weight-   k Specific heat ratio-   Z Fluid compressibility factor-   R₀ Universal gas constant-   ρ Density of fluid at the compressor inlet (or outlet)-   D Linear dimension of a compressor or piping characteristic-   P_(t_in) Total or stagnation pressure at compressor inlet-   T_(t_in) Total or stagnation temperature at compressor inlet-   P_(t_out) Total or stagnation pressure at compressor outlet-   T_(t_out) Total or stagnation temperature at compressor outlet

Where:

$\begin{matrix}{{V = \frac{4{\cdot m}}{\rho \cdot \pi \cdot D^{2}}}{V = {{V_{in}{and}\rho} = \rho_{in}}}} & (6)\end{matrix}$

if the flow meter located at the inlet, V=V_(out) and ρ=ρ_(out) if theflow meter located at the outlet; π is a mathematical constant ofapproximately 3.14; D is the diameter of the cross-section area atcompressor inlet (D_(in) ²) or outlet (D_(out) ²).

For compressor inlet:

$\begin{matrix}{\rho_{in} = \frac{{Mw} \cdot P_{t\_{in}}}{Z_{in} \cdot R_{0} \cdot T_{t\_{in}}}} & (7)\end{matrix}$ $\begin{matrix}{a_{in} = \sqrt{\frac{k_{in} \cdot R_{0} \cdot T_{t\_{in}}}{Mw}}} & (8)\end{matrix}$ $\begin{matrix}{k_{in} = \frac{C_{P}}{C_{v}}} & (9)\end{matrix}$

Mach number at compressor inlet:

$\begin{matrix}{\Pi_{1_{\_ in}} = \frac{V_{in}}{a_{in}}} & (10)\end{matrix}$

Mach number at compressor outlet:

$\begin{matrix}{\Pi_{1_{\_{out}}} = \frac{V_{out}}{a_{out}}} & (11)\end{matrix}$

Compressor pressure ratio (total to total):

$\begin{matrix}{\Pi_{2} = \frac{P_{t\_{out}}}{P_{t\_{in}}}} & (12)\end{matrix}$

And then:

$\begin{matrix}{T_{t\_{in}} = {T_{in} \cdot \left( {1 + {\frac{k - 1}{2} \cdot \Pi_{1{\_{in}}}^{2}}} \right)}} & (13)\end{matrix}$

where T_(in)—static temperature at the compressor inlet in absoluteunits.

$\begin{matrix}{T_{t\_{out}} = {T_{out} \cdot \left( {1 + {\frac{k - 1}{2} \cdot \Pi_{1{\_{out}}}^{2}}} \right)}} & (14)\end{matrix}$

where T_(out)—static temperature at the compressor outlet in absoluteunits.

For incompressible flow:

$\begin{matrix}{P_{t\_{in}} = {P_{in} \cdot \left( {1 + {\frac{k}{2} \cdot \Pi_{1{\_{in}}}^{2}}} \right)}} & (15)\end{matrix}$

where P_(in)—static pressure at the compressor inlet in absolute units,Π_(1_out)—Mach number at compressor inlet ≤0.3.

$\begin{matrix}{P_{t\_{out}} = {P_{out} \cdot \left( {1 + {\frac{k}{2} \cdot \Pi_{1{\_{out}}}^{2}}} \right)}} & (16)\end{matrix}$

where P_(out)—static pressure at the compressor outlet in absoluteunits, Π_(1_out)—Mach number at compressor outlet ≤0.3.

Whenever the Mach number in the stream exceeds about 0.3, the streambecomes compressible and the density of the fluid can no longer beconsidered as constant.

For compressible flow:

$\begin{matrix}{P_{t\_{in}} = {P_{in} \cdot \left( {1 + {\frac{k - 1}{2} \cdot \Pi_{1{\_{in}}}^{2}}} \right)^{\frac{k}{k - 1}}}} & (17)\end{matrix}$

where P_(in)—static pressure at the compressor inlet in absolute units,Π_(1_in)—Mach number at compressor inlet >0.3.

$\begin{matrix}{P_{t\_{out}} = {P_{out} \cdot \left( {1 + {\frac{k - 1}{2} \cdot \Pi_{1{\_{out}}}^{2}}} \right)^{\frac{k}{k - 1}}}} & (18)\end{matrix}$

where P_(out)—static pressure at the compressor outlet in absoluteunits, Π_(1_out)—Mach number at compressor outlet >0.3.

The relationship between the Mach numbers at the inlet and outlet of thecompressor, given that Z_(in)≅Z_(out) and k_(in)≅k_(out), follows fromthe equation:

$\begin{matrix}{\Pi_{1{\_{in}}} = {\frac{D_{out}^{2}}{D_{in}^{2}} \cdot \Pi_{1{\_{out}}} \cdot \left( \Pi_{2} \right)^{({1 - \frac{n - 1}{2 \cdot n}})}}} & (19)\end{matrix}$

Where n is the polytropic exponent, which can be calculated using theequation:

$\begin{matrix}{n = \left( {1 - \frac{\left( \frac{T_{t\_{out}}}{T_{t\_{in}}} \right)}{\left( \frac{P_{t\_{out}}}{P_{t\_{in}}} \right)}} \right)^{({- 1})}} & (20)\end{matrix}$

With a moderate change in friction in the system, n changesinsignificantly and can be taken in calculations as a constant.

In applications where differential pressure meters are used the inletMach number Π_(1_in) can be calculated from the equation:

$\begin{matrix}{\Pi_{1{\_{in}}} = {\frac{4 \cdot {Const}}{\pi \cdot D_{in}^{2} \cdot \sqrt{k_{in}}} \cdot \sqrt{\frac{\Delta P_{in}}{P_{in}}}}} & (21)\end{matrix}$

where ΔP_(in)—is the pressure drop across of the flow meter at the inletto the compressor, P_(in) is the static pressure at the compressor inletin absolute units, Const is the flow meter constant, π is a mathematicalconstant of approximately 3.14, D_(in)—internal diameter of the inletpipe.

Typical performance curves of dynamic variable speed compressors withoutguide vanes are shown in FIG. 2 in π-terms, with shape of the curvesvarying from slope to horizon for compressors with a nominal compressionratio of about 1.1 to 2.5, intermediate curves for compressors with anominal compression ratio of 2.5 to 6.0, and relatively straightvertical lines, for compressors with compression ratio higher than 6.0.It should be noted that compressors equipped with guide vanes can haverelatively vertical lines at a compression ratio of about 1.1 to 2.5with the blades closed. FIG. 3 displays a constant speed performancecurve in the π-term coordinates: Π_(1_in), is the Mach number atcompressor inlet, and (Π₂−1) is the compressor pressure ratio (total tototal) minus one; operating point between surge point A and choke pointB; rays emanating from the zero point, indicating possible movements ofthe operating point; r_(surge) is the radial coordinate and φ_(surge) isthe angular coordinate of the surge point; r_(choke) is the radialcoordinate and φ_(choke) is the angular coordinate of the choke point.The technique of converting the constant speed performance curve fromrectangular coordinates to polar coordinates is based on the assumptionthat the distance from the zero point to the surge point A and thedistance from zero point to the choke point B are equal. To equalize twounequal radial coordinates, a polar conversion factor P is entered. Thedistance from the zero point to the surge point A can then be calculatedusing the polar conversion factor:

r _(surge)=√{square root over ((Π₂−1)_(A) ²+(P·Π _(1_in))_(A) ²)}  (22)

where (Π₂−1)_(A) and (Π_(1_in))_(A)—coordinates of the surge point A.

The distance from the zero point to the choke point B can also becalculated using the polar conversion factor:

r _(choke)=√{square root over ((Π₂−1)_(B) ²+(P·Π _(1_in))_(B) ²)}  (23)

(Π₂−1)_(B) and (Π_(1_in))_(B)—coordinates of the choke point B.

From the two equations (22) and (23), assuming r_(surge)=r_(choke) thepolar conversion factor P for the AB constant speed performance curvecan be calculated as:

$\begin{matrix}{P = \sqrt{\frac{\left( {\Pi_{2} - 1} \right)_{A}^{2} - \left( {\Pi_{2} - 1} \right)_{B}^{2}}{\left( \Pi_{1{\_{in}}} \right)_{B}^{2} - \left( \Pi_{1{\_{in}}} \right)_{A}^{2}}}} & (24)\end{matrix}$

FIG. 4 shows a set of hypothetical constant speed performance curveswith points A₁, A₂, A₃ . . . A_(n−1), A_(n) and A_(n+1) as surge points;and points B₁, B₂, B₃ . . . . B_(n−1), B_(n) and B_(n+1) as chokepoints. Performance curves may differ from each another, but curvesformed by the same compressor, operating in a moderately narrow rangewith little change in system friction, should be correlated in shape.For this reason, the polar conversion factor P shouldn't change much.However, the polar conversion factor must be calculated for each givencurve shown in FIG. 4 . Then the arithmetic means or average of polarconversion factors, the sum of the polar conversion factors divided bytotal number of curves in the set (n+1), must be used to convert arectangular coordinate system in a polar coordinate system.

$\begin{matrix}{P_{{mean}\_{average}} = \frac{P_{1} + P_{2} + P_{3} + \ldots + P_{n - 1} + P_{n} + P_{n + 1}}{n + 1}} & (25)\end{matrix}$

In an imaginary two-dimension polar coordinate system on the plane, eachpoint corresponds to a pair of polar coordinates (r, φ). The operatingpoint, located on the constant speed curve A_(n)B_(n) as shown in FIG. 4, has a radial coordinate r_(op) and an angular coordinate φ_(op), whichis measured from the vertical axis (Π₂−1). An example of the polarconversion of the constant speed performance curves and positioning ofthe operating point on the A_(n)B_(n), curve of FIG. 4 is shown in FIG.5 , where the new modified constant speed curves and the operating pointon the line A_(n)B_(n) are shown in the polar coordinates plotted on therectangular coordinates with the vertical axis as the radial coordinater, and the horizontal axis as the angular coordinate φ. The polarcoordinates plotted on the rectangular coordinates, as shown in FIG. 5 ,illustrate the transformation effect, where the compressor performancecurves are straightened and can now be approximated by horizontal lines,and the shaded area surrounded by the surge and choke limiting linesdefines the dynamically stable area of the compressor operation.

The equations for calculating of a pair of polar coordinates (r, φ) foreach point are shown below:

$\begin{matrix}{r = \sqrt{\left( {\Pi_{2} - 1} \right)^{2} + \left( {P_{{mean}\_{average}} \cdot \Pi_{1{\_{in}}}} \right)^{2}}} & (26)\end{matrix}$ $\begin{matrix}{\varphi = {{ARCTAN}\left( \frac{P_{{mean}\_{average}} \cdot \Pi_{1{\_{in}}}}{\left( {\Pi_{2} - 1} \right)} \right)}} & (27)\end{matrix}$

Where ARCTAN is the inverse mathematical function of the tangentfunction used to obtain an angle from any of the trigonometric angularrelations.

The functions shown below in tabular form in TABLE 1 with sorted rowsand columns of characteristic data represent the polar angles of thesurge and choke points as functions of the radial coordinate r as anargument.

TABLE 1 Radial Polar angle of Polar angle of coordinate surge pointchoke point 1 (r)₁ (φ_(surge))₁ (φ_(choke))₁ INPUT (r_(op)) → 2 (r)₂(φ_(surge))₂ (φ_(choke))₂ 3 (r)₃ (φ_(surge))₃ (φ_(choke))₃ n − 1(r)_(n−1) (φ_(surge))_(n−1) (φ_(choke))_(n−1) n (r)_(n) (φ_(surge))_(n)(φ_(choke))_(n) n + 1 (r)_(n+1) (φ_(surge))_(n+1) (φ_(choke))_(n+1) ↓ ↓OUT1 (φ_(surge)) OUT2 (φ_(choke))

The definition of the functions is taken from FIG. 5 when the shadedarea is crossed by the horizontal lines r at the indicated surge andchoke points used to populate TABLE 1.

It should be noted that all points other than those inserted in the rowsand columns can be considered interpolated values. Linear interpolationis applied to a specific value between the two values listed in thetable, which can be achieved by geometric reconstruction of a straightline between two adjacent points in the table.

The use of table functions is that the input to the table is the radialcoordinate of the operating point r_(op) calculated from to the equation(26), and the outputs are the angular coordinates φ_(surge) andφ_(choke) of the surge and choke points. The graphical definition of thefunctions is shown in FIG. 5 where the shaded area crossed by thehorizontal line r_(op) and the angular coordinates φ_(surge) andφ_(choke) are the projections of the surge and choke points onto thehorizontal axis φ with the operating point φ_(op) between them as thecalculated value according to the equation (27). Therefore, thecontrolled variable CV (%) in percent for surge protection is calculatedas:

$\begin{matrix}{{{CV(\%)} = {100{\% \cdot \frac{\varphi_{op} - \varphi_{surge}}{\varphi_{choke} - \varphi_{surge}}}}}❘}_{r_{op}} & (28)\end{matrix}$

and for chock protection:

$\begin{matrix}{{{CV(\%)} = {100{\% \cdot \frac{\varphi_{choke} - \varphi_{op}}{\varphi_{choke} - \varphi_{surge}}}}}❘}_{r_{op}} & (29)\end{matrix}$

The shape of the constant speed performance curves can change fromcompressor to compressor or as the compressor operating range expands.However, the conversion method represented by equations (26) and (27) isapplicable to any shape of performance curve. The compressor performancecurves shown in FIG. 2 , which have significant shape variations, willstill be straightened when plotted in rectangular coordinates r versusφ, and the appearance of the modified constant speed performance curvescan be approximated by horizontal lines.

The generalized correlation between the controlled variable CV (%) inpercent and the polytropic efficiency of the compressor η_(p) in percentcovering entire operating range from surge to choke limits, is shown inFIG. 6 . The offset, between CV(η_(p_max))—the compressor's maximumpolytropic efficiency point and CV(η_(p_op)) of the operating point canbe used in PID controllers as a controlled variable to balance the loadbetween the compressors operating in parallel or in series.

FIG. 7 shows a schematic diagram of two dynamic compressors 14 and 15installed in series, the second compressor 15 having a side stream inletflow. The input signals shown in FIG. 7 : rotational speed 24, staticpressure at the inlet of the first compressor 16, static temperature atthe inlet of the first compressor 17, measured flow rate at the inlet ofthe first compressor 18, calculated mass flow rate of the firstcompressor 19, static inlet pressure of the second compressor 20, staticinlet temperature of the second compressor 21, measured side stream flowrate entering the second compressor 22, calculated mass flow of the sidestream entering the second compressor 23, static output pressure 25,static output temperature 26.

The total mass flow m_(total) through the second compressor 15 is thencalculated as the sum of the mass flow m₁ through the first compressor14 plus the side stream mass flow m₂ entering between compressors:

m _(total) =m ₁ +m ₂  (30)

To protect the second compressor, the Mach number (Π_(1_in))_(2_total)for the second stage must be used, which is calculated from the totalmass flow m_(total), assuming that this mass flow passes through theinlet of the second compressor. For differential pressure meters, takinginto account that compressibility factors and specific heat ratios ofthe first and second compressors are equal Z₁≅Z₂ and k₁≅k₂ the Machnumber (Π_(1_in))_(2_total) can be calculated as:

$\begin{matrix}{\left( \Pi_{1{\_{in}}} \right)_{2{\_{total}}} = {{\frac{D_{1}^{2}}{D_{2}^{2}} \cdot \left( \Pi_{2} \right)_{1}^{({\frac{n - 1}{2 \cdot n} - 1})} \cdot \left( \Pi_{1{\_{in}}} \right)_{1}} + \left( \Pi_{1{\_{in}}} \right)_{2}}} & (31)\end{matrix}$

Where D₁ is the diameter of the cross-section area at the inlet of thefirst compressor; D₂—cross-section diameter at the inlet of the secondcompressor; (Π₂)₁—pressure ratio across the first compressor, calculatedaccording to equation (12); n is the polytropic exponent of the firstcompressor, it can be taken as a constant or calculated by equation(20); (Π_(1_in))₁—Mach number at the inlet of the first compressor and(Π_(1_in))₂—Mach number of the side stream of the second compressor,both calculated according to equation (21).

In many cases, variable geometry compressors with the IGV inlet guidevanes or stator vanes in axial compressors are used. Compressors of thistype can have performance drift depending on the blades opening. Theeffect of IGV opening on the compressor performance is shown in FIG. 8for three hypothetical sets of constant speed performance curvesrepresenting three arbitrary selected IGV opening positions 0%, 50% and100% in coordinates Π_(1_in) relative to (Π₂−1). Each set consists offour curves for demonstrative purposes. Lines A₁B₁, A₂B₂, A₃B₃ and A₄B₄represent constant speed performance curves at 0% IGV position; linesA′₁B′₁, A₂′B′₂, A′₃B′₃ and A′₄B′₄ represent constant speed performancecurves at 50% IGV position; and lines A″₁B″₁, A″₂B″₂, A″₃B″₃ and A″₄B″₄represent constant speed performance curves at 100% IGV position.

FIG. 8 graphically illustrates a technique for adapting three separatesurge limiting lines of three different IGV positions into one commonsurge line. Points A₃, A′₃ and A″₃ in the FIG. 8 refer to the samecompressor speed selected as the assumed design operating speed. Thepoints are chosen as an example to obtain the function ƒ(IGV) of theinlet guide vanes to modify the n-term coordinate of the Mach number.The shift of the surge points A₃, A′₃ and A″₃ to the left while keepingtheir (Π₂−1) coordinates unchanged, denotes the new positions of thesurge points A₃com, A′₃com and A″₃com, which form one common surge line.

Dividing the coordinates of the surge points A₃com, A′₃com and A″₃cominto the values of the coordinates of the surge points A₃, A′₃ and A″₃,respectively, reveals the method for constructing the IGV function:

$\begin{matrix}{{f\left( {IGV} \right)} = \frac{\left( \Pi_{1{\_{in}}} \right)_{A\_{com}}}{\left( \Pi_{1{\_{in}}} \right)_{A}}} & (32)\end{matrix}$

FIG. 9 shows the position of the inlet guide vanes IGV as a percentagerelative to the IGV function and depicts the technique, for calculatingthe IGV function. The function shown below in tabular form in TABLE 2with two columns of characteristic data for an IGV position as theargument from 0% to 100% and the function ƒ(IGV) of all available surgepoints of design operating speed to plot the expected common surge line.

TABLE 2 IGV position function % ƒ(IGV) 1  0%$\frac{\left( \prod_{1{\_{in}}} \right)_{0\%{\_{com}}}}{\left( \prod_{1{\_{in}}} \right)_{0\%}}$INPUT (IGV) → 2  10%$\frac{\left( \prod_{1{\_{in}}} \right)_{10\%{\_{com}}}}{\left( \prod_{1{\_{in}}} \right)_{10\%}}$ 50%$\frac{\left( \prod_{1{\_{in}}} \right)_{50\%{\_{com}}}}{\left( \prod_{1{\_{in}}} \right)_{50\%}}$i-1  90%$\frac{\left( \prod_{1{\_{in}}} \right)_{90\%{\_{com}}}}{\left( \prod_{1{\_{in}}} \right)_{90\%}}$i 100%$\frac{\left( \prod_{1{\_{in}}} \right)_{100\%{\_{com}}}}{\left( \prod_{1{\_{in}}} \right)_{100\%}}$↓ OUTPUT (ƒ(IGV)

The result of applying the inlet guide vanes function to three sets ofconstant speed performance curves for three IGV opening positions inFIG. 8 is presented in FIG. 10 , which shows one combined set of allconstant speed, performance curves in the ƒ(IGV)·Π_(1_in) if coordinaterelative to the (Π₂−1) coordinate. Where all surge points can beapproximated by a single surge line, and also all choke points can bealigned to form a choke line.

The same method of converting constant speed performance curves fromrectangular to polar coordinates can now be applied to compressors withIGVs, provided that the π-term coordinate Π_(1_in) is replaced by thenew coordinate ƒ(IGV)·Π_(1_in). An equal distance statement stating thatthe distance from the zero point to the surge point A, and the distancefrom zero point to the choke point B for each performance curve in FIG.10 is still required for coordinate conversion. To equalize the twounequal radial coordinates of the surge and choke points, it is alsonecessary to calculate the polar conversion factor P.

The distance from the zero point to each surge point A can then becalculated as:

r _(surge)=√{square root over ((Π₂−1)_(A) ²+(P·ƒ(IGV)·Π_(1_in))_(A)²)}  (33)

where (Π₂−1)_(A) and (ƒ(IGV)·Π_(1_in))_(A)—coordinates of the surgepoints A.

The distance from the zero point to each choke point B can be calculatedas:

r _(choke)=√{square root over ((Π₂−1)_(B) ²+(P·ƒ(IGV)·Π_(1_in))_(B)²)}  (34)

where (Π₂−1)_(B) and (ƒ(IGV)·Π_(1_in))_(B)—coordinates of the chokepoints B.

From the two equations (33) and (34), by assigning r_(surge)=r_(choke),the polar conversion factor P for each constant speed performance curveAB can be calculated as:

$\begin{matrix}{P = \sqrt{\frac{\left( {\Pi_{2} - 1} \right)_{A}^{2} - \left( {\Pi_{2} - 1} \right)_{B}^{2}}{\left( {{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)_{B}^{2} - \left( {{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)_{A}^{2}}}} & (35)\end{matrix}$

After the polar conversion factors have been calculated for each curve,it is necessary to calculate the arithmetic means or average of thepolar conversion factors, the sum of the polar conversion factorsdivided by the total number of curves in the sets (m+1):

$\begin{matrix}{P_{{mean}\_{average}} = \frac{P_{1} + P_{2} + P_{3} + \ldots + P_{m - 1} + P_{m} + P_{m + 1}}{m + 1}} & (36)\end{matrix}$

FIG. 11 illustrates the transformation effect, where the polarcoordinates r and φ are plotted again in rectangular coordinates withstraightened compressor performance curves, and the shaded area boundedby the surge and choke limiting lines defines the compressor operatingarea. The operating point in FIG. 11 is defined by the radial coordinater_(op) and the angular coordinate φ_(op). As before in this invention,each point in the two-dimension polar coordinate system on the plane hasa pair of polar coordinates (r, φ), but the equations for calculatingthe polar coordinates (r, φ) with respect to the IGV function areadjusted as shown below:

$\begin{matrix}{r = \sqrt{\left( {\Pi_{2} - 1} \right)^{2} + \left( {P_{{mean}\_{average}} \cdot {f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)^{2}}} & (37)\end{matrix}$ $\begin{matrix}{\varphi = {{ARCTAN}\left( \frac{P_{{mean}\_{average}} \cdot {f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}}{\left( {\Pi_{2} - 1} \right)} \right)}} & (38)\end{matrix}$

TABLE 1 can now be filled with surge and choke points taken from FIG. 11. After calculating the radial coordinate r_(op) of the operating point,the angular coordinates to φ_(surge) and φ_(choke) can be obtained fromTABLE 1. This is graphically shown in FIG. 11 with the projections ofthe surge point φ_(surge) and the choke point φ_(choke) on thehorizontal axis φ, and the calculated angular coordinates φ_(op) of theoperating point between them. The controlled variable CV (%) in percentcan then be calculated for surge protection using equation (28), and forchoke protection from equation (29).

A hypothetical compressor map is shown in FIG. 12 in π-term coordinatesΠ_(1_in) and Π₂−1) without choke line, where A points are still thesurge points and B points are the maximum flow endpoints on eachperformance curve. In this case, when the full range of compressoroperation, defined from surge limit to choke limit, is not available,the controlled variable CV (%) can only be calculated for the surgeprotection.

The rays emanating from the zero point in FIG. 12 indicate the angularcoordinates of all surge points from the polar angle α_A₁ of the firstsurge point A₁ to the polar angle α_A_(n+1) of the last surge pointA_(n+1), Then FIG. 13 shows modified compressor map, where each surgepoint on the surge line now has the same angular coordinate. This isachieved by replacing the π-term Mach number coordinate Π_(1_in), withthe coordinate (Π_(1_in))_(Corr), which is the corrected Mach number asa function of the π-term (Π₂−1) obtained from surge points by theformula:

(Π_(1_in))_(A)=(Π₂−1)_(A)  (39)

The function shown below in tabular form in TABLE 3 with two columns ofcharacteristic data, where (Π_(1_in))_(A) is the argument and (Π₂−1)_(A)is the function derived from FIG. 13 for surge points.

TABLE 3 π-term Mach (Π₂ − 1) number as function 1 (Π₁_in)_(A) ₁ (Π₂ −1)_(A) ₁ INPUT (Π₁_in) → 2 (Π₁_in)_(A) ₂ (Π₂ − 1)_(A) ₂ 3 (Π₁_in)_(A) ₃(Π₂ − 1)_(A) ₃ n − 1 (Π₁_in)_(A) _(n-1) (Π₂ − 1)_(A) _(n-1) n(Π₁_in)_(A) _(n) (Π₂ − 1)_(A) _(n) n + 1 (Π₁_in)_(A) _(n+1) (Π₂ − 1)_(A)_(n+1) ↓ OUTPUT (Π₁_in)_(Corr)

The same technique of converting constant speed performance curves fromrectangular to polar coordinates can now be applied to compressors withonly the surge limit line, provided that the n-term coordinate Π_(1_in)is replaced by the new coordinate (Π_(1_in))_(Corr).

To equalize the two unequal radial coordinates of the surge and maximumflow endpoint, it is also necessary to calculate the polar conversionfactor P.

The distance from the zero point to each surge point A can then becalculated as:

r _(surge)=√{square root over ((Π₂−1)_(A) ²+(P·(Π_(1_in))_(Corr))_(A)²)}  (40)

where (Π₂−1)_(A) and ((Π_(1_in))_(Corr))_(A)—coordinates of the surgepoints A.

The distance from the zero point to each maximum flow endpoint B can becalculated as:

r _(max_flow)=√{square root over ((Π₂−1)_(B) ²+(P·(Π_(1_in))_(Corr))_(B)²)}  (41)

where (Π₂−1)_(B) and ((Π_(1_in))_(Corr))_(B)—coordinates of the maximumflow points B.

From the two equations (40) and (41), setting thatr_(surge)=r_(max_flow), the polar conversion factor P for each ABconstant speed performance curve can be calculated as:

$\begin{matrix}{P = \sqrt{\frac{\left( {\Pi_{2} - 1} \right)_{A}^{2} - \left( {\Pi_{2} - 1} \right)_{B}^{2}}{\left( \left( \Pi_{1{\_{in}}} \right)_{Corr} \right)_{B}^{2} - \left( \left( \Pi_{1{\_{in}}} \right)_{Corr} \right)_{A}^{2}}}} & (42)\end{matrix}$

The arithmetic means or average of the polar conversion factors, the sumof the polar conversion factors divided by the total number of curvescan be calculated using the equation (25).

FIG. 14 illustrates the effect of polar transformation by displaying aradial coordinate r and an angular coordinate α in rectangularcoordinates. FIG. 14 shows the straitened performance curves and theshaded area bounded by a surge line on one side, defined as a verticalline with a constant angular coordinate, and a line, connecting theendpoints on the other side.

The equations for calculating of a pair of polar coordinates (r, α) areshown below:

$\begin{matrix}{r = \sqrt{\left( {\Pi_{2} - 1} \right)^{2} + \left( {P_{{mean}\_{average}} \cdot \left( \Pi_{1{\_{in}}} \right)_{Corr}} \right)^{2}}} & (43)\end{matrix}$ $\begin{matrix}{\alpha = {{ARCTAN}\left( \frac{P_{{mean}\_{average}} \cdot \left( \Pi_{1{\_{in}}} \right)_{Corr}}{\left( {\Pi_{2} - 1} \right)} \right)}} & (44)\end{matrix}$

TABLE 4 is populated with surge points and maximum flow endpoints takenfrom FIG. 14 , where the surge points polar angle column is constant.After calculating the radial coordinate r_(op) of the operating point,the angular coordinates α_(const) and α_(max_flow) are obtained fromTABLE 4.

TABLE 4 Radial Polar angle of Polar angle of coordinate surge pointmaximum flow point 1 (r)₁ α_(const) (α_(max)_flow)₁ INPUT (r_(op)) → 2(r)₂ α_(const) (α_(max)_flow)₂ 3 (r)₃ α_(const) (α_(max)_flow)₃ n − 1(r)_(n-1) α_(const) (α_(max)_flow)_(n-1) n (r)_(n) α_(const)(α_(max)_flow)_(n) n + 1 (r)_(n+1) α_(const) (α_(max)_flow)_(n+1) ↓ ↓OUT1 (α_(const)) OUT2 (α_(max)_flow)

Graphically it is shown in FIG. 14 with the constant coordinate of thesurge points (angular coordinate α_(const)) and the projection of themaximum flow endpoint (angular coordinates α_(max_flow)) onto thehorizontal a axis with the calculated operating point (angularcoordinates α_(op)) between them.

The controlled variable CV (%) in percent for the surge protectioncontroller in the case of maximum flow endpoints can be calculatedrelative to the surge limit as the polar angle of the operating pointα_(op) minus the constant α_(const) (polar angle of the surge points)divided by the specified operating range up to maximum flow line,defined as subtracting the constant from the polar angle of the maximumflow endpoint α_(max_flow):

$\begin{matrix}{{{CV(\%)} = {100{\% \cdot \frac{\alpha_{op} - \alpha_{const}}{\alpha_{\max\_{flow}} - \alpha_{const}}}}}❘}_{r_{op}} & (45)\end{matrix}$

It can be assumed that the hypothetical compressor map, shown in FIG. 12in π-term coordinates Π_(1_in) and (Π₂−1) has only surge points Aobtained during commissioning. The rays emanating from the zero point inFIG. 12 still indicate the angular coordinates of all surge points, fromthe polar angle α_A₁ of the surge point A₁ to the polar angle α_A_(n+1)of the surge point A_(n+1). Similarly, the surge line and surge pointsmay be presented in a corrected Mach number coordinate (Π_(1_in))_(Corr)as the function of n-term coordinates (Π₂−1) in FIG. 13 , where eachsurge point on the surge line has the same polar angle. The correctedMach number coordinate (Π_(1_in))_(Corr) still can be found from theequation (38), and TABLE 3 with two columns of characteristic dataobtained from surge points is still applicable when only surge pointsare available. In the absence of performance curves, the calculation ofthe polar conversion factor P would be impossible. A pair of polarcoordinates (r, α) can be calculated from equation (43) and (44) withthe parameter P_(mean_average) equal to one:

$\begin{matrix}{r = \sqrt{\left( {\Pi_{2} - 1} \right)^{2} + \left( \left( \Pi_{1{\_{in}}} \right)_{Corr} \right)^{2}}} & (46)\end{matrix}$ $\begin{matrix}{\alpha = {{ARCTAN}\left( \frac{\left( \Pi_{1{\_{in}}} \right)_{Corr}}{\left( {\Pi_{2} - 1} \right)} \right)}} & (47)\end{matrix}$

The controlled variable CV (%) in percent for the surge protectioncontroller can be calculated as the polar angle of the operating pointα_(op) minus constant α_(const) the polar angle of the surge points,divided by the polar angle of the surge points:

$\begin{matrix}{{CV(\%)} = {100{\% \cdot \frac{\alpha_{op} - \alpha_{const}}{\alpha_{const}}}}} & (48)\end{matrix}$

If surge points are collected during commissioning with a flow meterlocated downstream of the compressor, the n-term Mach number iscalculated as the Mach number at the outlet of the compressor. FIG. 15shows surge line and surge points A in the π-term coordinates Π_(1_out)and (Π₂−1), as they are obtained from field tests without performancecurves. Very often, for a Mach number calculated at the outlet of thecompressor, starting from a nominal compression ratio of about 4.0 to5.0 and above, the surge line can become vertical. In this case, thereare two ways to calculate the percentage controlled variable CV (%). Thefirst uses equation (19), which links the Mach numbers at the compressorinlet and outlet by changing Π_(1_out) to Π_(1_in).

The second uses the π-term coordinate Π_(1_out), but the π-termcoordinate (Π₂−1) is replaced with a new corrected coordinate so thateach surge point has the same polar angle. This is achieved by replacingthe π-term coordinate (Π₂−1) with the coordinate (Π₂−1)_(Corr), which isa function of the π-term Mach number Π_(1_out) obtained from surgepoints by the formula:

(Π₂−1)_(A)=(Π_(1_out))_(A)  (49)

FIG. 15 shows a surge line with angular coordinates of all surge pointsfrom the polar angle γ_A₁ of the first surge point A₁ to the polar angleγ_A_(n+1) of the last surge point A_(n+1). FIG. 16 shows the modifiedsurge line in rectangular coordinates, but with surge points having thesame polar angle or the polar coordinate γ_(const). The function shownbelow in tabular form in TABLE 5 with two columns of characteristicdata, where the coordinate (Π₂−1)_(A) is the argument and(Π_(1_out))_(A) is the function, are obtained from surge points in FIG.15 .

In the absence of compressor characteristic curves, the polar radius rcan be calculated from the equation below:

r=√{square root over (((Π₂−1)_(Corr))²+(Π_(1_out))²)}  (50)

and the angular coordinate γ can be calculated using the equation:

$\begin{matrix}{\gamma = {{ARCTAN}\left( \frac{\Pi_{1{\_{out}}}}{\left( {\Pi_{2} - 1} \right)_{Corr}} \right)}} & (51)\end{matrix}$

The controlled variable CV (%) in percent for the surge protectioncontroller can be calculated as the polar angle of the operating pointγ_(op), minus constant γ_(const) the polar angle of the surge points,divided by the polar angle of the surge points:

$\begin{matrix}{{CV(\%)} = {100{\% \cdot \frac{\gamma_{op} - \gamma_{const}}{\gamma_{const}}}}} & (52)\end{matrix}$

TABLE 5 (Π₁_out) π-term (Π₂ − 1) as function 1 (Π₂ − 1)_(A) ₁(Π₁_out)_(A) ₁ 2 (Π₂ − 1)_(A) ₂ (Π₁_out)_(A) ₂ INPUT (Π₂ − 1) → 3 (Π₂ −1)_(A) ₃ (Π₁_out)_(A) ₃ n − 1 (Π₂ − 1)_(A) _(n−1) (Π₁_out)_(A) _(n−1) n(Π₂ − 1)_(A) _(n) (Π₁_out)_(A) _(n) n + 1 (Π₂ − 1)_(A) _(n+1)(Π₁_out)_(A) _(n+1) ↓ OUTPUT ((Π₂ − 1)_(Corr))

The effect of the IGV opening on compressor performance is shown in FIG.17 , similar to that shown in FIG. 8 , with the difference that B pointsrepresent the endpoints of the maximum flow. Therefore, FIG. 17represents a case where a hypothetical compressor map is shown inTc-teen coordinates Π_(1_in) and (Π₂−1) with surge line but no chokeline; with three sets of constant speed performance curves representingthree IGV opening positions of 0%, 50% and 100%, and each of themconsists of four curves. Likewise, lines A₁B₁, A₂B₂, A₃B₃ and A₄B₄ referto constant speed performance curves at 0% IGV position; lines A′₁B′₁,A₂′B′₂, A′₃B′₃ and A′₄B′₄ to constant speed performance curves at 50%IGV; and lines A″₁B″₁, A″₂B″₂, A″₃B″₃ and A′₄B″₄ for constant speedperformance curves at 100% IGV. Since the surge points are identical tothose shown in FIG. 8 , and do not change their positions, the IGVfunction shown in TABLE 2 can be used.

FIG. 18 shows the result of applying the inlet guide vanes function tothree sets of constant speed performance curves with one common surgeline in coordinates ƒ(IGV)·Π_(1_in) and (Π₂−1). Again, in the absence ofa choke line, the control variable CV (%) can only be calculated forsurge protection. The rays emanating from the zero point in FIG. 18indicate the angular coordinates of all surge points from the polarangle α_A₁ corn of the surge point A₁com to the polar angle α_A″₄com ofthe surge point A″₄com.

FIG. 19 shows a modified compressor map, which is a modification of thecompressor map shown in FIG. 18 , where the π-term coordinate (Π₂−1) isreplaced with a new corrected coordinate such that each surge point hasthe same polar angular. This is achieved in the same way as before,replacing the π-term coordinate (Π₂−1) with the coordinate(Π₂−1)_(Corr), but as a function of the π-term Mach numberƒ(IGV)·Π_(1_in) of the surge points shown in FIG. 18 . The correctedcoordinate (Π₂−1)_(Corr) is calculated for each surge point using theformula:

(Π₂−1)_(A)=(ƒ(IGV)·Π_(1_in))_(A)  (53)

The same method of converting constant speed performance curves fromrectangular to polar coordinates can now be applied to compressors withthe IGV and the endpoints of the maximum flow. An equal distancestatement for each performance curve that declares the distance from thezero point to the surge point A and from the zero point to the maximumflow endpoint B, as well as the calculation of the polar conversionfactor P, are still required for polar conversion.

The distance from the zero point to each surge point A can then becalculated as:

r _(surge)=√{square root over ((Π₂−1)_(Corr))_(A)²+(P·∫(IGV)·Π_(1_in))_(A) ²)}  (54)

where ((Π₂−1)_(Corr))_(A) and (ƒ(IGV)·Π_(1_in))_(A)—coordinates of thesurge points A.

The distance from the zero point to each maximum flow endpoint B can becalculated as:

r _(max_flow)=√{square root over ((Π₂−1)_(Corr))_(B)²+(P·∫(IGV)·Π_(1_in))_(B) ²)}  (55)

where ((Π₂−1)_(Corr))_(B) and (ƒ(IGV)·Π_(1_in))_(B)—coordinates of themaximum flow points B.

From the two equations (54) and (55), setting r_(surge)=r_(max_flow),the polar conversion factor P for each AB constant speed performancecurve can be calculated as:

$\begin{matrix}{P = \sqrt{\frac{\left( \left( {\Pi_{2} - 1} \right)_{Corr} \right)_{A}^{2} - \left( \left( {\Pi_{2} - 1} \right)_{Corr} \right)_{B}^{2}}{\left( {{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)_{B}^{2} - \left( {{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)_{A}^{2}}}} & (56)\end{matrix}$

The arithmetic mean P_(mean_average) can be calculated from the formula(36) as the sum of the polar conversion factors divided by the totalnumber of curves in the sets. As before, in a two-dimension polarcoordinate system on the plane, each point corresponds to a pair ofpolar coordinates (r, α), but equations for calculating the polarcoordinates (r, α) must be adjusted as shown below:

$\begin{matrix}{r = \sqrt{\left( \left( {\Pi_{2} - 1} \right)_{Corr} \right)^{2} + \left( {P_{{mean}\_{average}} \cdot {f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)^{2}}} & (57)\end{matrix}$ $\begin{matrix}{\alpha = {{ARCTAN}\left( \frac{P_{{mean}\_{average}} \cdot {f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}}{\left( {\Pi_{2} - 1} \right)_{Corr}} \right)}} & (58)\end{matrix}$

FIG. 20 illustrates a polar transformation, where the polar coordinatesr and α are plotted in rectangular coordinates, the performance curvesare flattened, and the shaded area surrounded by the surge limiting lineand the maximum flow line, defines the compressor operating area.

TABLE 6 is populated with surge points and maximum flow endpoints takenfrom FIG. 20 , where the polar angle of the surge points is constant,and the polar angles of the maximum flow endpoints marked with a symbol(▪) are defined as the points of intersection of the r coordinates withthe maximum flow line at points (B)₁, (B)₂, (B)₃ . . . (B)_(n−1),(B)_(n) and (B)_(n+1).

The radial coordinates r_(op) and α_(op) of the operating point can becalculated from equations (57) and (58). The angular coordinatesα_(const) and α_(max_flow) are obtained from TABLE 6.

TABLE 6 Radial Polar angle Polar angle of coordinate of surge pointmaximum flow point 1 (r)₁ α_(const) (α_(max)_flow)_((B)1) 2 (r)₂α_(const) (α_(max)_flow)_((B)2) INPUT (r_(op)) → 3 (r)₃ α_(const)(α_(max)_flow)_((B)3) n − 1 (r)_(n-1) α_(const) (α_(max)_flow)_((B)n−1)N (r)_(n) α_(const) (α_(max)_flow)_((B)n) n + 1 (r)_(n+1) α_(const)(α_(max)_flow)_((B)n+1) ↓ ↓ OUT1 (α_(const)) OUT2 (α_(max)_flow)

Graphically it is shown in FIG. 20 with the constant surge pointcoordinate (angular coordinates α_(const)) and, the projection of themaximum flow endpoint (angular coordinates α_(max_flow)) onto thehorizontal axis α with the calculated operating point (angularcoordinates α_(op)) between them. The controlled variable CV (%) inpercent for the surge protection controller in the case of a variablegeometry compressor with surge line and maximum flow endpoints can becalculated relative to the surge limit from equation (45).

It can now be assumed that only surge points A in FIG. 17 are known. Inthe same way, the displacement of the surge points A₃, A′₃ and A″₃ tothe left with unchanged (Π₂−1) coordinates denote the new positions ofthe surge points A₃com, A′₃com and A″₃com on the line defined as theexpected common surge line. The IGV function can still be calculated bydividing the coordinates of the surge points A₃com, A′₃com and A″₃com bythe coordinates of the surge points A₃, A′₃ and A″₃, respectively. AndTABLE 2 again can be populated with characteristic data representing theIGV position from 0% to 100%, and a function ƒ(IGV) obtained for allavailable surge points by forming the expected common surge line.

In absence of the performance curves the rays emanating from the zeropoint in FIG. 18 still indicate the angular coordinates of all surgepoints from the polar angle α_A₁ com of the surge point A₁com to thepolar angle α_A″₄com of the surge point A″₄com. Surge points A stillhave the equal polar angle in the π-term coordinates ƒ(IGV)·Π_(1_in) and(Π₂−1)_(Corr) in FIG. 19 . In the case when only surge points arepresent, the polar radius r can be calculated from equation (57) and theangular coordinate α can be calculated from equation (58), where theparameter P_(mean_average) is equal to one:

$\begin{matrix}{r = \sqrt{\left( \left( {\Pi_{2} - 1} \right)_{Corr} \right)^{2} + \left( {{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}} \right)^{2}}} & (59)\end{matrix}$ $\begin{matrix}{\alpha = {{ARCTAN}\left( \frac{{f\left( {IGV} \right)} \cdot \Pi_{1{\_{in}}}}{\left( {\Pi_{2} - 1} \right)_{Corr}} \right)}} & (60)\end{matrix}$

And then the controlled variable CV (%) in percent for the surgeprotection controller can be calculated from equation (48).

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 19. A method forcontrolling the operation of a centrifugal or axial compressor equippedwith automatic control systems that continuously calculate systemparameters, said method comprising: reading one or more input signalsfrom one or more sensors; converting a compressor performance mapcomprising one or more surge points which define a first boundarycondition into a compressor flow function in rectangular coordinates offlow Mach number and total pressure ratio; applying said compressor flowfunction to each of said one or more surge points; converting said oneor more surge points of said first boundary condition to polarcoordinates at a constant angle; measuring an operating point of thecentrifugal or axial compressor via said input signals from said one ormore sensors; calculating a control variable in polar coordinates;calculating an error value from a difference between a set point andsaid control variable in polar coordinates; and sending a control signalto a compressor control mechanism such that said control variable ismoved closer to said set point to reduce said error value.
 20. Themethod of claim 19, wherein said compressor flow function comprises atotal pressure ratio function; and applying said total pressure ratiofunction to each of said one or more surge points to define an alteredcoordinate of flow Mach number.
 21. The method of claim 19, wherein saidcompressor flow function comprises a flow Mach number function; andapplying said flow Mach number function to each of said one or moresurge points to define an altered coordinate of total pressure ratio.22. The method of claim 19, wherein said compressor control mechanismcomprises a mechanism selected from the group consisting of ananti-surge valve and an outlet valve.
 23. A method for controlling theoperation of a centrifugal or axial compressor equipped with automaticcontrol systems that continuously calculate system parameters, saidmethod comprising: reading one or more input signals from one or moresensors; converting a compressor performance map comprising one or morechoke points which define a second boundary condition into a compressorflow function in rectangular coordinates of flow Mach number and totalpressure ratio; applying said compressor flow function to each of saidone or more choke points; converting said one or more choke points ofsaid second boundary condition to polar coordinates at a constant angle;measuring an operating point of the centrifugal or axial compressor viasaid input signals from said one or more sensors; calculating a controlvariable in polar coordinates; calculating an error value from adifference between a set point and said control variable in polarcoordinates; and sending a control signal to a compressor controlmechanism such that said control variable is moved closer to said setpoint to reduce said error value.
 24. The method of claim 23, whereinsaid compressor flow function comprises a total pressure ratio function;and applying said total pressure ratio function to each of said one ormore choke points to define an altered coordinate of flow Mach number.25. The method of claim 23, wherein said compressor flow functioncomprises a flow Mach number function; and applying said flow Machnumber function to each of said one or more choke points to define analtered coordinate of total pressure ratio.
 26. The method of claim 23,wherein said compressor control mechanism comprises a mechanism selectedfrom the group consisting of an outlet valve, a variable inlet guidevane controller, and a variable stator vane controller.